Question

Need a little help setting up an integral:

Answer 1

Consider \(\large \int_{C} (x^2+y^3)dx+3xy^2dy \) where C is the parabola \(y=x^2\) from (2,4) to (0,0) followed by the line segment from (0, 0) to (2, 0) and another line from (2,0) to (2, 4)

Evaluate the integral using Green's Theorem

Answer 2

@hartnn

Answer 3

could you plot the region ?

Answer 4

\(\Large \oint_{C} (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, dx\, dy\)

Answer 5

something really weird happens with that integrand thingy
doesn't it?

Answer 6

Yes, I end up getting 0 when taking the partials, unless I did that wrong... I had set it from y: 0 -> 2x and x: 0 -> 2

Answer 7

nah I got the same thing for the integrand

Answer 8

oh, thats right, it becomes 0

Answer 9

That's why I got confused.

Answer 10

I never applied green's theorem before...
but if I were to read this like a "normal" integral we are looking at nothing

Answer 11

you know like when I think about this integral I think about something like:
\[\int\limits_{1}^{2}0 dx=0\]
this 0 because

Answer 12

but @hartnn as I said I'm not familar with green's theorem
are there any conditions that the integrand must satisfy?

Answer 13

"If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there ..."
~Wiki

Answer 14

so i guess, there are no conditions other than that you should be able to find partial derivatives,
in this case, you could find it...so you can apply

Answer 15

So using Green's Theorem would technically give me the same answer as if I solved it using line integrals?

Answer 16

thats right...can we find the answer using line integral and verify it to be =0 ... ?

Answer 17

I don't know how to set up the integral as a line integral though, not used to the form :/

Could you help me set it up? :P